P and Q are 2 distinct points in the interior of the square ABCD such that AP =Aq and angle BAP is congruent to angle DAQ.Show that APCQ is a KITE
To show it's a kite, you need to show that AP = AQ and PC = QC.
That's because a kite has two pairs of adjacent sides that are the same.
You already have that AP = AQ, all you need to do then is show PC = QC, which can be done with straightforward geometry. Draw it and you'll find the figure you have is mirror-symmetrical in the diagonal AC.